## The Problem

Imagine you have a list of possible questions you want students to be able to understand and be able to translate into mathematical symbols, like the following.

1. The sum of six and a number
2. Eight more than a number
3. A number plus five
4. A number increased by seven
5. Seven more than a number
6. The difference of five and a number
7. Four less than a number
8. Seven minus a number
9. A number decreased by nine
10. The product of nine and a number

One common approach I have seen used is to model a few of the questions on the list and then ask students to attempt the other problems themselves or in small groups. This approach has a serious flaw; it requires students to know the thing you are trying to teach in order to do the task.

Let’s imagine a very similar task, except now suppose I ask you to translate the list into German. Here are the first two phrases translated into German (thank you anonymous translators).

1. Die Summe von sechs und eine Zahl.
2. Acht mehr als eine Zahl.

Now translate the other 8 sentences.

Unless you already know German, you can’t do this task. Worse, imagine I gave you the entire list in German and asked you to translate it into Hebrew.

1. Die Summe von sechs und eine Zahl.
2. Acht mehr als eine Zahl.
3. Eine Zahl plus fuenf.
4. Eine Zahl von sieben vermehrt.
5. Sieben mehr als eine Zahl.
6. Der Unterschied von fuenf und einer Zahl.
7. Vier weniger als eine Zahl.
8. Sieben minus eine Zahl.
9. Eine Zahl verringert bei neun.
10. Das Produkt von neun und eine Zahl.

Here are the first two phrases translated into Hebrew. Translate the rest of the phrases from German.

1. החיבור של שש ומספר
2. שמונה יותר ממספר

Unless you know German well enough to understand the phrases in the first place and Hebrew will enough to translate the German, you cannot do this task. You also cannot do this task if you do not know how these phrases are related to each other in the two different languages.

If your students do not know the vocabulary in the first list I shared and/or they do not know the mathematical symbols, then they cannot do the translation between the two without some intervention.

## A solution:

However, students may be able to use their partial knowledge of the symbols or the vocabulary to fill in gaps in either. As an alternative to having them work on the entire list from scratch you could:

• Give them a copy of both lists and ask them to individually match as many of the items as possible and then attempt to match the other ones as best as they can. They could then work in groups to refine and improve their lists and then individually try to write out the situations using actual numbers.
• Embed the different phrases in a context through which students could make sense of the relationships between the symbols and the text. One issue here with making sense of what these phrases mean is that students do not have sufficient clues as to what they could mean because the phrases are completely contextless. There is not enough information provided as to what these phrases actually mean. Here’s an example of a task based on equality instead of inequalities but hopefully you can generalize what I mean from it.

If students have insufficient knowledge of either the vocabulary or the mathematical symbols, then they need to build that knowledge first. In this case, these ten recommendations on building vocabulary may be useful to consider.

The original goal is probably not a very good goal given that students are rarely, if ever, asked to translate phrases this short into mathematical terminology. Instead of focusing on the small building blocks students might use to translate phrases, it is more useful to start with longer phrases based on meaningful contexts (note: this does not necessarily mean real world) that include more text and to work with students to reduce these phrases to simplest form, and then use these reduced forms to look for mathematical connections between the longer forms of text.

• One problem I have frequently encountered, especially with engineering students (whose mathematical abilities are quite reasonable), is the inability to translate back from the symbolic to the verbal and meaningful.  Ask a student what y = mx + c says and you get “why equals em ex plus sea”. I need say no more, except that with simple linear differential equations it is no better.

ps. Just as well I made a copy of this, as any error in the submission procedure deletes the message.

• David Wees wrote:

In this case, my suspicion is that the engineering students either do not know what the mathematical symbols mean or they lack experiences connecting the mathematical symbols back to any contexts. They probably understand the contexts themselves were they explained in their spoken/written language.

ps. I’ll see if I can a better comment submission process.

• Peter Smyth wrote:

iNow that I have four grandkids all seven or younger, I’ve gotten interested in how kids acquire vocabulary and meaning. One striking thing is that how we approach it in school is probably not a “best practice”. The second thing is that we are wired to do this. I happens naturally if we let it happen.

I think it grows out of context, and kids make sense of situations and connect the words and meaning the the present in terms of past experiences. The idea of “more than ” comes before the words and certainly the symbols.

So maybe we need to think of this thing of attaching words to ideas that they already get or are coming to understand. Words as containers for ideas.

• I agree entirely. School rushes into symbols way too fast. I think the same goes for fractions and ratios. Kids have a good idea of “I’ve only got half as much as her!”. “Our car goes twice as fast as yours”. And so on, long before “fractions”. I have very strong ideas on this.

Soon I will do a post on this subject.