Math Makes Sense 3 Answer Key Pdf

[Berenguer Miramontes]

Asa fan of CGI, I know children are naturally sense-makers. But I also know that reading mathematical problems is a special kind of reading, and students need instruction in it. Historically, teachers have used two different types of instruction for reading word problems:

A fix for overuse of NEI is to get those who choose it to either:

(a) explain what information they would like to have to answer the question; or

(b) show that there are multiple answers consistent with the information that is given.

To be clear, I am not saying these are desirable characteristics or that we should be emphasizing these in our classes, but I think they are the most common type of question in math classes and math exams. Of course, MCQs appear in other settings, too, particularly riddles/puzzles.

Finally, I will ask my children at home this question and I am sure they will give me numerical answers. The real test is how much they are giggling when they do so, because they love to give me silly answers, especially when they think the question is silly.

Your list of 5 traits of MCQs is really interesting. Thank you! What was interesting to me was to see a kindergartener and first-grade students following the cues, even without much classroom experience. How did they get trained up on them?

Looking across thousands of student responses, the most common answers are: 9, 8, 132, 29, 27, 8.25, and 11. One of the interesting things here is that not a single one of those answers is computationally incorrect. This begs the question, how did students come up with these answers? Can we figure all of them out?

So, we can figure out a bit about how each individual student is making sense of the problem and where they could use some support. But another way is to look at how students who answer a certain way are performing in general.

Here we can see a quantitative difference between the answers. Combining that with the student responses above, we can tell that there are different levels of sense making here. I would say that the answer 9 is making sense of the question context and units (students divided by students per table equals tables) and they got it computationally correct.

I believe this explains the difference in state test scores between students getting the answer 132 and 29 or 37. I would also imagine we would see a higher percentage of students who wrote 8.25 using division than we would see in students who submitted the answer 8. So, within sense groups, different tools will contribute to the level of sophisticated computational thinking students have and thus will relate to their state test scores.

With the idea of highlighting sense-making, I tagged 16 of our assessments, (each with about 8-12 questions) with the ways that students made sense of a problem. In each case, I was careful to make sure that sense-making was tagged to an incorrect answer as well to make sure I was thinking through it properly.

For 36, students made sense of physical space, including the sophisticated step of utilizing the information given to find the missing side length to include in the perimeter. However, it does not make sense of the context of the problem regarding the mathematical language of area. Still, it is computationally correct. 35 is the same, but is computed incorrectly.

Contextual Numbers: This includes answers that answered a different question than was asked. Quick example: What is the total price you would pay for an $80 shirt that was on sale for 40% off. An answer of $32 is correct computationally and in units, but is contextually inaccurate as that is the amount of the discount as opposed to the price you are paying.

Patterns: Patterns includes students correctly identifying a growth pattern or other patterns in the questions asked of them. For the time being, I have included different types of relationships inside of patterns. For example, correctly identifying a proportional relationship vs. a linear relationship vs. an exponential relationship.

Parameters: I did not include this in the questions I tagged, but this would include mapping the parameters of a function to the numbers they represent. For example, given the slope and intercept, could you write an equation in the form y = mx + b. Popular equations could include the pythagoream theorem, area and volume formulas for different shapes, trigonometry, etc.

After analyzing all these assessments and giving each student a metric on their sense making for all of the different types above, I looked at their correlation to the state test scores. We see a strong correlation between the individual types of conceptual sense-making, and when combined together, these become highly correlated to state test results. This was done with tens of thousands of students over multiple semesters and school districts.

Now what in math? Just yesterday, I visited a fifth grade classroom where a student calculated that 1/3 + 7/12 was worth 7/36, while his partner across the table said the sum was 11/12. When I asked about the discrepancy, the first student offered that they might be both correct, as these could be equivalent fractions. He did not cry. He was not concerned. He had no expectation that the math ought to make any sense whatsoever.

When students memorize only procedural skill in mathematics, they miss the backstory, they miss the opportunity to question their own calculations, they miss the meaning: What do these numbers represent? How are we combining them? What would be a reasonable solution?

Making Sense of Math

In order to support students believing not only that math makes sense but that they themselves can be sense-makers as mathematicians, we need to slow down, ask deeper questions, offer and analyze models, , and practice transfer.

Without understanding concepts, students might still master procedures, even produce proficient scores on assessments. Yet my young friend yesterday demonstrated the folly of knowing procedures without knowing the why: he handily and accurately multiplied two fractions instead of adding them, as the problem prompted. But because the meaning was lost on him, the discrepancy between 7/36 and 11/12 was of no concern. He even had a possible explanation: equivalent fractions.

Understanding is as important in math as it is in reading. When we slow down, talk in depth, model for meaning and practice transferring ideas into alternate contexts, we encourage all learners to believe that math is meaningful and that they themselves are meaning-makers. We can teach students to want mathematical meaning so bad they could cry.

So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an enjoyable and rewarding experience. However, I have come to realise how much further I still have to go to reach a point where I could even think about publishing original contributions in the literature given how intensively everything has already been studied and the discoveries already made.

For example, I have just finished a 600 page textbook on graduate level mathematics. Although it took me a while to understand everything in it, I learned from this and enjoyed doing the exercises, but realised by the end that I still basically know nothing and that it is really intended as a springboard to slightly more advanced texts. I picked up another book which starts to delve more into one of the specific aspects in the book and again, it is 500 pages long.

Later: I am reading this a few years later and realise the question could be hard to answer, as depends on many things (there are some problems where one could contribute decisively without knowing any math at all). However, I will leave the question as I think it's something that many students ask themselves and there is some useful generic advice in the answers.

To be sure, there are some subfields of mathematics that are highly technical, and you're unlikely to be able to contribute something new to them unless you've studied a lot of background material. However, there are also areas of mathematics that don't require that much background knowledge. For example, Aubrey de Grey recently made spectacular progress on a longstanding open problem in combinatorics, and almost no background knowledge was needed for that problem. Even in supposedly highly technical areas of mathematics, people sometimes come up with breakthroughs that employ very little advanced machinery.

As others have mentioned, more crucial than "knowing everything" are (1) finding a good problem to work on, and (2) having problem-solving ability. If you have both of these, then you can typically learn what you need as you go along. When you're at an early stage in your career, finding a good problem generally requires an advisor, unless you have the rare ability to smell out good problems yourself just by reading the literature and listening to talks. Problem-solving ability is probably innate to some extent, but a lot of it comes down to experience and persistence. Of course you will be a more powerful problem solver if you have a lot of tools in your toolbox, but generally speaking, you get better at solving problems by spending your time directly attempting to solve problems, and only reading the 500-page books when it becomes clear that they are needed to solve the problem you have in mind.

Mathematics is not learned by reading books. One becomes a research mathematician by solving problems. Most people need an adviser to recommend a good problem. Then you start thinking and reading what is relevant to your specific problem. General education by reading books with hundreds of pages can be done as a parallel process, but the main emphasis should be on a specific problem. It is a duty of the adviser to find a problem which does not require too much reading.

There are many examples that demonstrate these principles. Many good mathematicians obtained their first original results before the age of 18 or even much earlier,at the time whenthey learned very little.

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