Here's something that percolates through K-9 math education:
Always relate your projects, exercises, and tests to real-life and the real-world. Teach students that math is necessary in the real-world. Students want to know how math applies to their world. They want to see its practical value. Doing this will make it appealing to them.
Well, duh! Looks like an obvious way to make math more appealing. But things "gang aft agley".
Consider this scenario: You are making up a question for elementary students that requires multidigit addition. So you come up with:
Ageing Relatives
(Also known as "When in doubt, add!")
John’s aunt is 35, Tom’s uncle is 42. What is their combined age?
Without the trigger word "combined" why would you even think of adding the ages?
Young children try to make sense of new experiences. Unfortunately, there is not much opportunity for sense-making with this problem.
The setting (
two people of different ages) is real, but the context (
the necessity for doing the math) is missing. To solve the
Ageing Relatives problem, children are obliged to do math in a setting that does not provide a rationale for actually doing the math. Give them too many problems like this one, and they may start operating according to this principle:
When there are two numbers and it is not clear how to use them,
the proper thing to do is to add them.
There is evidence that this happens. A while back, this problem swept through the web:
Robert Kaplinsky gave the question to 32 grade eight students, and 24 of his students came up with a numerical answer. (You can see
his discussion here along with reactions from other teachers.) It is not clear whether the students provided numerical answers because they wanted to please the teacher, or whether they acted reflexively and just did some arithmetic with the numbers that happened to be available.
Here's an algebraic variant that my daughter sent me:
Here is a more advanced version of the "math without rationale" syndrome.
Training a Mouse
(A magic function problem)
Alex has been training a mouse to find a reward by carrying out a certain task. He wants to know how long it will take the mouse the find the reward on its fifth try.
The time that it takes the mouse to find the reward is modelled by
T(n) = 0.04n2 – 6n + 30,
where
T is the time in seconds and
n is the number of times the mouse has previously tried the task.
How long will it take the mouse to find the reward on its fifth try?
A
magic function is one with a unfathomable connection to the setting, but which nevertheless must be used to obtain a solution for the problem. Since the derivation of the magic function is unknowable, it carries with it this message:
Any useful parts of mathematics will forever remain beyond our understanding.
In truth, the sole intent of a magic function problem is to have the student do something with the given function –– in this particular example, it is to evaluate the function when
n = 4. In problems like this, the function may or may not be realistic.
I don't know if magic function problems can be fixed without scrapping the function, in which case one should discard the setting and leave the question as a procedural exercise.
Below are a few more vexatious real-world problems. I tried to show how to fix the problem by providing alternate settings that still retain the math task while avoiding the awkwardness of the original setting. Some of the fixes employ a setting that is not real-life. (I believe that most students will accept a pretend-world context as long
as it does not attempt to pass itself off as their world.)
The Diner Menu
(Math is everywhere?)
Andrew saw the following sign at a diner. If he bought one of each item and spent $7.50, how much did the drink cost?
| ITEM | COST ($) |
|---|
| Burger | 3x + 0.05 |
| Fries | x |
| Drink | x + 0.10 |
The question is nice and short. But have you ever seen product prices listed like this?
Here's a possible fix that uses a fictitious setting.
(I'll agree with you that the fix is not much better than the original question, but at least it removes that incredibly bogus real-world setting.)
On the forest moon of the planet Endor there lived an Ewok family with 3 children. The parents had given their children some Baubles and Bangles. Here's what each child has:
| Child | Collection |
|---|
| Infant Ewok | 1 Bauble |
| Pre-school Ewok | 1 Bauble & 2 Bangles |
| Teen Ewok | 3 Baubles & 1 Bangle |
The children want to exchange their Baubles for Bangles, and the parents have agreed to do this.
Unfortunately, none of the Ewoks could remember how many Bangles a Bauble was worth. All they could remember was that, taken all together, the totality of the children's Baubles and Bangles had the same value as 18 Bangles.
After the exchange, how many Bangles will each Ewok child have?
The following is a different fix, also not real-world, and it's a bit too wordy, but the setting does provide a plausible reason for the way the data is presented.
Mrs Boychuk and her two young daughters were visiting an historical tower that had three observation levels. Leading between the levels were one or more
red staircases sometimes followed by
extra wooden steps.
All of the red staircases were identical. Mrs Boychuk asked Freddy to count the number of steps in a red staircase, and asked Frankie to count the number of wooden steps between each level.
Freddy dashed as fast as she could to the top level and enthusiastically reported that there were
90 steps in all. In her exuberance, she didn't count the number of steps in a red staircase.
This is the data they collected:
| Levels | Red stairs | Wooden stairs |
|---|
| Level 0 to Level 1 | 1 staircase | 0 |
| Level 1 to Level 2 | 1 staircase | 10 |
| Level 2 to Level 3 | 3 staircases | 5 |
Of course, Mrs Boychuk was a bit peeved, since she was going to ask the youngsters to add in columns and find the total number of steps. But then she realized that she had a new problem to ask:
How many steps are there between each observation level?
Cookie Boxes
("The cart before the horse it is!")
Before her coffee break, Ms Kowalchuk had prepared 24 boxes of cookies, and tied each box with a red ribbon. When she came back from her coffee break she noticed that the ribbons were removed from 4 boxes and 7 boxes were missing.
How many boxes with red ribbons were left?
Many of us have had a lot of experience with word problems. When I see one like this, I tend to react reflexively and proceed to solve it without considering the reasonableness of the problem.
How did Ms Kowalchuk know that 7 boxes were missing
without first counting how many boxes were left and subtracting that from 24? In other words, the answer to the question had to be known before there was enough information to ask the question.
We can make the math task seem more reasonable by changing things so that the setting itself
explains
how Ms Kowalchuk noticed that 7 boxes were missing without already knowing the answer she is seeking.
Before her coffee break, Ms Kowalchuk had prepared treats for her community league book club. She put out 24 plates each with a cupcake and two cookies. When she came back, she saw immediately that four plates only had cookies on them, and she counted seven plates that were completely empty.
How many plates were left that still had a cupcake?
Apple Inventory
(Using pliers to hammer a nail?)
A grocery store manager noted that on Tuesday, for every 3 McIntosh apples the store sold, they sold 4 Red Delicious apples. On Tuesday they sold 24 McIntosh apples.
How many Red Delicious apples did they sell?
My friend is an excellent handyman. He vents when he sees someone using the wrong tools. For me, using ratio and proportion to track inventory also calls for a vent. It is a situation where an inappropriate mathematical tool is being used to do a real-world task for which other tools are available and would certainly be used.
As in the previous problem, a change in the setting will fix things and still retain the desired math.
A recipe for making 3 quiches calls for 4 eggs. A pastry shop wants to prepare 24 quiches using that recipe.
How many eggs will they need to make 24 quiches?
The Cargo Truck
(Blind spots.)
Use the following information to answer the question below
A total of 10 packages are arranged in the back of a cargo truck as shown in the diagram below. One large package has the same mass as two medium packages. One medium package has the same mass as three small packages.
Question
How many small packages need to be loaded onto the right side of the truck to balance the load?
A. 8
B. 9
C. 12
D. 13
I had more fun with this problem than any of the others because it illustrates one of the hazards of imposing a real-world setting. The setting has blind spots –– properties that are extraneous to the math task, but which provide alternate and more reasonable approaches to solving the actual real world problem. And when this happens it can turn a good math task into an annoying one.
For the Cargo Truck problem, the real-world setting is a cargo truck loaded with packages of differing masses.
The real-world problem is to balance the load. The
solution method is imposed by whoever set the problem:
add more small boxes to the load.
In real-life, the most obvious way to fix an imbalance would be to reposition the packages already in the truck. (And if that were permitted, then there is an answer where no extra packages are needed.)
Also, a real-life cargo company would know the masses of the individual packages rather than their masses relative to each other. Knowing the individual masses would lead to a different approach to the problem. (Also, this raises the "cart before the horse" objection: it is not clear how the company would obtain the relative masses without first knowing the actual masses of each package.)
The diagram itself could be misleading for some students. It suggests that there is not enough space for the extra small packages that are required as ballast. (Thirteen are required unless you reposition some packages, but we have already discussed that).
On the other hand,
the imposed solution method involves some interesting math, and it worth finding a setting, real-world or not, that works.
Here are two different fixes. This first one retains the notion of balance:
There are three different types of boxes. All green boxes weigh the same, all blue boxes weigh the same, and all orange boxes weigh the same.
|
|
| These scales balance. |
These scales balance. |
How many more orange boxes are needed on the right pan to make the scales balance?
(You cannot move the boxes on the left.)
The second modification drops the demand that the setting must be 100% real-world. This setting uses the concept of money, so it should be familiar to the students, yet different enough to signal that the question is not trying to pass itself off as truly real-world.
A country uses three different coloured coins. Green ones are called buckazoids. Blue ones are called halfzoids. Orange ones are called mini-zoids.
| Each buckazoid is worth two halfzoids. |
 |
| Each halfzoid is worth three minizoids. |
 |
Mary and John each have the money shown:
|
|
| Mary's money. |
John's money. |
Their mother has promised to give John enough minizoids so that he has the same amount of money as Mary.
How many minizoids should their mother give to John?
Am I against using real-world contexts? No. but, but sometimes they can produce really bizarre questions. Paradoxically, extensive use of real-world settings can cause a dissociation between math and the real world.
For more on this, see
this post by
Nat Banting and
this Globe and Mail column written by
Sunil Singh a few years ago.
Problem Sources
The Diner Menu is due to
Cathy Yenca, who posted the question in the sequence of articles by
Dan Meyer about
pseudocontext in math problems. There are many more examples in those articles.
The Cargo Truck is a question from the
sample grade 6 Alberta PAT. Alberta Education considers this question to be illustrative of the more challenging ones in the PAT.
The other problems are rewordings, paraphrases, or amalgams of questions that I found on worksheets and other resources posted on the web.
