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Thinking mathematically: Man vs. Machine

11 10 2009 

For the lesson summary click through.

This activity sheet started with quick summary of my calculators in math research; their ubiquitous nature, parents’ fears, an example, and research that shows calculators do not harm skill development if used properly. The front page also contained a clear statement of students learning goals for this lessons. Highlight this so students read this first and get context for this lesson. Instead of reading the large text I skimmed over it for them, asking some questions during that (Who likes calculators? Who thinks you don’t learn as much with them? Who uses them when out shopping?).

The bottom of the first page outlined the lesson sequence or “Today’s tasks”. Students read the first task and then set about to it. They had to fill out a T-chart on the “benefits” and “downfalls” of using a calculator. Here students are told if they are stuck for ideas, they should try reading the first page text in detail as it has some ideas. Students work on the chart as individuals for 60 seconds and then in pairs for 90 seconds. At this stage each pair contributes to a chart on the board. I did half the pairs giving a positive and the other half negative, but you could alternate or another method. Try to clarify vague answers such as using a calculator for “big” sums – do they mean large numbers, or many numbers, or many different operations?

Students then return to their task list, if you feel like being tricky ask them which step they are up to. Steps 2 and 3 were actually the pairwork and group discussion respectively, so they are actually up to step 4 – the quiz. This should have them all excited. The quiz could work as individuals, pairwork or groups. I think groups allows for more comfortable balance of competitiveness and cooperativeness. Try to put students in random or mixed-ability groups to provide balance. Set some sort of callsign or buzzer technique for students to use when they have the answer.

The questions should be a mix of those that students can do without a calculator (basic number facts), those that a calculator is helpful (adding long lists of numbers, subtracting large numbers, multiplying 2- or 3-digit numbers), and those that a calculator is pretty much essential (division by 3-digit numbers or greater, very long lists of numbers). Try to include mostly those in the first two categories, which will allow students to recognise that calculators are actually only vital in certain situations.

Also add some “thinking” and “hosejob” questions. “Thinking” should mean more complex problems. One I used was a shopping list; students had to work out the change from $50, two groups gave the total of the items, not the change (they also forgot to include units, units are important!). Thinking may also include fill-in-the-blanks, such as 55 x 5 = ? x 11; a few students gave the answer of 55 x 5 rather than what the question asked (perhaps that should be counted as a hosejob). The best hosejob is to have a complex sum such as 452 x 1899 + 2456 ÷ 16 which most students will probably use a calculator for, on the next card have 2456 ÷ 16 + 1899 x 452, the exact same problem with the exact same answer. Some students will see the link, some will be able to once it is explained. Another good question is to add or subtract whole millions, students can learn that “big numbers” take a long time to put into a calculator; you don’t need a calculator to do: 5,000,000 + 3,000,000 – 2,000,000 + 500.

As students answer questions I asked them to explain how they got their answer. Students will be hesitant on how to phrase this, so help a few word them. Instead of saying “I used a calculator”, students should be saying “I put that exact sum into the calculator” or “I did the first part in a calculator and then doubled it”. It is important for students to realise you can do some problems in different ways. For the shopping list question you could add all the numbers, and then subtract this from 50; or you could start with 50 and subtract each number in turn; personally, I would add all the numbers and subtract 50, and then change the resulting negative number into a positive.

Reflection can also continue after the quiz to recognise how they might improve their T-bars further now. I then asked students to complete a personal reflection on what they learned today and hand it up (the last part of the activity sheet).

My students responded quite well to this lesson. I worked with 12 students at a time (4 groups of 3 for the quiz – randomly assigned by colouring in the “learning goals” box on their sheets). I think the quiz might get unwieldly with the whole class, students won’t work together as well in larger groups, and distanced groups are at a disadvantage of reading questions (always read questions out when showing them). I probably should have done the pre-quiz activity as a whole class and then the quiz in two groups (students not in the quiz worked on assignments). You may also want students not involved in the quiz in another area to those who are; even though I turned desks around for students not in the quiz, some astute students tried to remember answers they heard being called out.

Other things to look out for running this activity:

Image credit: He’s going to do math for me by firepile from flickr (CC by A)

*I’d put the sheet up, but not all the images are free. Maybe after the Mayan apocalypse I’ll have an umbrella website I can host that sort of stuff.





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5 11 2009
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