MTEL Math Elementary (68) common mistakes
MTEL Math Elementary (68) · Understand the traps and how to avoid them.
When dividing a fraction by a fraction, the candidate inverts the dividend instead of the divisor, or flips both fractions, producing answers like (3/4)÷(1/2)=(3/4)×(1/2)=3/8 instead of 3/2.
The procedure is memorized as a chant ('flip and multiply') without attaching the 'flip' to a specific fraction, and the candidate has no mental model of why division by a number smaller than 1 yields a larger quotient. Under time pressure the chant runs on autopilot.
What to do instead
Rebuild the rule from a model the candidate can reproduce: rewrite (3/4)÷(1/2) as 'how many halves fit in three-quarters?' and check with a common-denominator argument, (3/4)÷(1/2)=(6/8)÷(4/8)=6÷4=3/2. Practice ten mixed problems where the answer must be predicted as larger or smaller than the dividend before computing, so a flipped divisor is caught immediately.
The candidate adds or subtracts fractions by operating on numerators and denominators separately, writing 2/3+1/4=3/7, and does not notice that the result is smaller than one of the addends.
Whole-number intuition is carried over: with whole numbers, adding the parts adds the total. The candidate never internalized that the denominator names the size of the piece, so pieces of different sizes cannot be counted together.
What to do instead
Use fraction strips or number-line jumps to show that thirds and fourths are different-sized units, then convert to twelfths before adding. Institute a habit check: any sum of two positive fractions must exceed each addend, and any difference must be smaller than the starting number. Apply the check to every fraction computation during practice until it is automatic.
The candidate converts between decimals, fractions, and percents by moving the decimal point the wrong number of places, for example writing 0.4% as 0.4 or 4 rather than 0.004, and cannot order 0.35, 3/8, and 34% correctly.
Place-value language ('percent means per hundred') is recited but not applied, and the candidate treats the percent sign as decoration rather than as an instruction to divide by 100. Mixed-format ordering tasks expose the gap because no single procedure covers them.
What to do instead
Drill the conversion as a two-step written routine: percent to decimal means divide by 100 (move two places left, so 0.4% becomes 0.004), decimal to percent means multiply by 100. For ordering tasks, convert every value to a decimal to three places on scratch paper before comparing. Verify with a magnitude check: 0.4% is less than 1%, so the answer must be tiny.
The candidate evaluates an expression such as 8+4×3÷2−1 strictly left to right, or computes 2+3² as 25, and reports an answer that ignores order-of-operations conventions.
Reading direction dominates learned rules, and exponents are misread as 'multiply the base by the exponent.' The candidate also does not recognize that grouping symbols override the standard order.
What to do instead
Have the candidate annotate every expression before computing: bracket each exponent, then each multiplication/division pair, then additions and subtractions, and only then calculate. Contrast 2+3²=11 with (2+3)²=25 and with 2×3²=18 to separate base from exponent. Practice with a calculator that honors order of operations so the candidate can self-correct.
Asked to find the rule for a pattern such as 3, 7, 15, 31, the candidate extends it by a constant difference (adding 4, then 8, then 16 is ignored) and writes the next term as 39 instead of 63.
The candidate checks only the first pair of terms and stops, assuming linearity. Elementary pattern work often rewards simple addition, so the candidate has not developed the habit of testing a rule against the whole sequence.
What to do instead
Require a written rule that reproduces every given term, not just the next one: 'double and add one' gives 3→7→15→31→63. Then translate the verbal rule into an expression for the nth term (2ⁿ⁺¹−1) and test it at n=1 and n=4. Practice sequences that mix linear, doubling, and squaring rules so the candidate must discriminate rather than default.
The candidate solves 3x+5=20 by subtracting 5 from the left side only, or by dividing before undoing the addition, and accepts x=15/3 written as 5 without checking that the original equation holds.
Inverse operations are applied as isolated steps rather than as a balance model, and the equals sign is read as 'the answer comes next' rather than as a statement of equality between two sides.
What to do instead
Model each equation on a balance: whatever is done to one side must be done to the other, and undo operations in reverse order (subtract 5 from both sides, then divide both sides by 3). End every problem by substituting the value back into the original equation and computing both sides. Do five equations per session with the substitution check written out, not skipped.
The candidate computes the area of a triangle as base×height (giving 48 instead of 24 for base 8 and height 6) or confuses area and perimeter formulas, reporting perimeter in square units.
Rectangle formulas are overlearned and transferred without adjustment, and the candidate does not connect the triangle formula to the fact that a triangle is half of a parallelogram. Units are treated as labels rather than as quantities.
What to do instead
Derive the triangle formula by drawing a rectangle, splitting it along a diagonal, and observing that each triangle is half. Then practice a mixed set where each problem asks first for the quantity (area or perimeter) and its unit before any computation. Enforce unit discipline: length in cm, area in cm², and reject any answer whose unit does not match the question.
The candidate computes the volume of a rectangular prism as length+width+height or as 2(lw+lh+wh) when volume is asked, and computes surface area when volume is asked, because the two formulas are recalled as one blurred cluster.
Volume and surface area are taught close together and both involve three dimensions, so the candidate stores them without a physical referent. No mental image distinguishes 'space inside' from 'skin outside.'
What to do instead
Anchor each formula to a physical object: fill a box with unit cubes to see volume as layers (l×w per layer, h layers), then wrap the same box to see surface area as six faces. Practice alternating questions on one shape ('how much water fits' versus 'how much paint covers') and require the candidate to name the quantity before choosing a formula.
The candidate reads a histogram or bar graph by comparing bar heights without accounting for unequal class widths or different totals, and reports that group A has 'more' when only its raw count is larger but its proportion is smaller.
The candidate treats all graphs as pictographs where taller always means more, and has not been asked to compare parts to wholes across groups of different sizes.
What to do instead
For every graph, have the candidate write the total for each group and convert counts to percentages before comparing. For histograms, check whether class widths are equal and compute frequency density where they are not. Practice with two groups of different totals (for example 30 of 50 versus 45 of 100) so that raw-count reasoning visibly fails.
The candidate computes the mean of a data set by averaging only the distinct values (for 2, 2, 2, 5, 9, adding 2+5+9 and dividing by 3), or reports the median of an even-sized set as one of the middle values instead of their average.
The candidate confuses 'average' with 'typical value' and collapses repeated entries, and the median procedure is memorized for odd counts only, so even counts have no working rule.
What to do instead
Enforce a written procedure: list every data point including repeats, count n, sum all n values, divide by n. For the median, sort the list, mark n, and if n is even average the two central values. Use a small set with a repeated value and an even count (2, 2, 2, 5, 9, 10) and compute mean, median, and mode side by side so the candidate sees how each treats repeats differently.